The dimension of sparse and co-sparse random graph orders
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| Format: | Preprint |
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2025
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| _version_ | 1866911394457190400 |
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| author | Gao, Pu Kumar, Arnav |
| author_facet | Gao, Pu Kumar, Arnav |
| contents | A random graph order is a partial order obtained from a random graph on $[n]$ by taking the transitive closure of the adjacency relation. The dimension of the random graph orders from random bipartite graphs $B(n,n,p)$ and from $G(n,p)$ were previously studied when $p=Ω(\log n/n)$ and when $p$ is not too close to 1. There is a conjectured phase transition in the sparse range at $p=1/n$. In this paper, we investigate this conjectured phase transition and estimate the dimension of the partial orders arising from $B(n,n,p)$ and $G(n,p)$ when $p=O(1/n)$. For the random bipartite order, we additionally estimate its dimension in the co-sparse regime, thereby closing all previously open ranges of $p$. Finally, we establish a general upper bound on the dimension of partial orders based on their decompositions into suborders, a result that is of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_19029 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The dimension of sparse and co-sparse random graph orders Gao, Pu Kumar, Arnav Combinatorics A random graph order is a partial order obtained from a random graph on $[n]$ by taking the transitive closure of the adjacency relation. The dimension of the random graph orders from random bipartite graphs $B(n,n,p)$ and from $G(n,p)$ were previously studied when $p=Ω(\log n/n)$ and when $p$ is not too close to 1. There is a conjectured phase transition in the sparse range at $p=1/n$. In this paper, we investigate this conjectured phase transition and estimate the dimension of the partial orders arising from $B(n,n,p)$ and $G(n,p)$ when $p=O(1/n)$. For the random bipartite order, we additionally estimate its dimension in the co-sparse regime, thereby closing all previously open ranges of $p$. Finally, we establish a general upper bound on the dimension of partial orders based on their decompositions into suborders, a result that is of independent interest. |
| title | The dimension of sparse and co-sparse random graph orders |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.19029 |